Laplace transform tables

A ready-to-use table of Laplace transforms, with properties and notable cases

laplace

transform

Key concepts

The Laplace transform is a powerful integral transform used extensively in mathematics, engineering, and physics to analyze linear time-invariant systems. Named after the French mathematician Pierre-Simon Laplace, this transform converts complex differential equations, which are often difficult to solve directly, into simpler algebraic equations. By transforming functions of time (typically denoted as t) into functions of a complex variable (denoted as s), the Laplace transform facilitates the analysis and solution of linear differential equations, making it an invaluable tool in both theoretical and applied contexts. Key properties and applications of Laplace transform are:

Linear Differential Equations: One of the primary uses of the Laplace transform is to solve linear differential equations with constant coefficients. By transforming these equations into the s-domain, their solution is greatly simplified.
Initial and Boundary Value Problems: The Laplace transform is particularly useful in solving initial value problems and boundary value problems, as it can incorporate initial conditions directly into the transformed equation.
Control Systems: In control engineering, the Laplace transform is used to analyze and design control systems. Transfer functions, which characterize system behaviour, are conveniently expressed in the s-domain.
Signal Processing: The Laplace transform aids in the analysis and manipulation of signals. It provides insights into the frequency components and stability of systems.
Electrical Circuits: In electrical engineering, the Laplace transform is used to analyze and solve circuit equations, especially when dealing with circuits involving capacitors and inductors.

Laplace transform definition

When not explicitly specified, the unilateral Laplace transform is assumed, which is defined as $$F(s) = \int_0^\infty f(t)e^{-st}dt \quad with \quad s=\sigma+i\omega \quad \sigma,\omega \in \mathbb{R}$$

Laplace transform properties

Expression		Equivalent	Expression		Equivalent
$$\mathcal{L}\left\{ a f(t) + b g(t) \right\} $$	$$\Rightarrow$$	$$a\mathcal{L}\left\{ f(t) \right\} + b\mathcal{L}\left\{ g(t) \right\} $$	$$\mathcal{L}\left\{ \int_0^t f(\tau) d\tau \right\} $$	$$\Rightarrow$$	$$\frac{F(s)}{s}$$
$$\mathcal{L}\left\{ \frac{df(t)}{dt} \right\} $$	$$\Rightarrow$$	$$sF(s)-f(0)$$	$$\mathcal{L}\left\{ \frac{d^2f(t)}{dt^2} \right\} $$	$$\Rightarrow$$	$$s^2F(s)-sf(0)-f'(0)$$
$$\mathcal{L}\left\{ f(t-a)u(t-a) \right\} $$	$$\Rightarrow$$	$$e^{-as}F(s)$$	$$\mathcal{L}\left\{ e^{at}f(t) \right\} $$	$$\Rightarrow$$	$$F(s-a)$$

Laplace transform table

$$\mathcal{L}^{-1}(F(s))$$	$$\mathcal{L}(f(t))$$	$$\mathcal{L}^{-1}(F(s))$$	$$\mathcal{L}(f(t))$$
$$1$$	$$\frac{1}{s}$$	$$e^{at}$$	$$\frac{1}{s-a}$$
$$t^n, n\in \mathbb N ^{+}$$	$$\frac{n!}{s^{n+1}}$$	$$t^p, p>-1$$	$$\frac{\Gamma (p+1)}{s^{p+1}}$$
$$\sqrt{t}$$	$$\frac{\sqrt{\pi}}{2s^{\frac{3}{2}}}$$	$$t^{n-\frac{1}{2}}, n\in \mathbb N ^{+}$$	$$\frac{\prod_1^n (2n-1) \sqrt{\pi}}{2^n s^{n+\frac{1}{2}}}$$
$$sin(at)$$	$$\frac{a}{a^2+s^2}$$	$$cos(at)$$	$$\frac{s}{a^2+s^2}$$
$$tsin(at)$$	$$\frac{2as}{(a^2+s^2)^2}$$	$$tcos(at)$$	$$\frac{s^2-a^2}{(a^2-s^2)^2}$$
$$sin(at+b)$$	$$\frac{s sin(b)+a cos(b)}{a^2+s^2}$$	$$cos(at+b)$$	$$\frac{s cos(b)+a sin(b)}{a^2+s^2}$$
$$sin(at)-at cos(at)$$	$$\frac{2a^3}{(a^2+s^2)^2}$$	$$sin(at)+at cos(at)$$	$$\frac{2as^2}{(a^2+s^2)^2}$$
$$cos(at)-at sin(at)$$	$$\frac{s(s^2-a^2)}{(a^2+s^2)^2}$$	$$cos(at)+at sin(at)$$	$$\frac{s(s^2+3a^2)}{(a^2+s^2)^2}$$
$$sinh(at)$$	$$\frac{a}{s^2-a^2}$$	$$cosh(at)$$	$$\frac{s}{s^2-a^2}$$
$$e^{at}sin(bt)$$	$$\frac{b}{(s-a)^2+b^2}$$	$$e^{at}cos(bt)$$	$$\frac{s-a}{(s-a)^2+b^2}$$
$$e^{at}sinh(bt)$$	$$\frac{b}{(s-a)^2-b^2}$$	$$e^{at}cosh(bt)$$	$$\frac{s-a}{(s-a)^2-b^2}$$
$$t^ne^{at}, n\in \mathbb N ^{+}$$	$$\frac{n!}{(s-a)^{n+1}}$$	$$f(at)$$	$$\frac{1}{a}F(\frac{s}{a})$$
$$u(t-a)$$	$$\frac{e^{-as}}{s}$$	$$\delta(t-a)$$	$$e^{-as}$$
$$u(t-a)f(t-a)$$	$$ e^{-as}F(s)$$	$$u(t-a)f(t)$$	$$e^{-as}\mathcal{L}(f(t+a))$$
$$e^{at}f(t)$$	$$F(s-a)$$	$$t^nf(t), n\in \mathbb N ^{+}$$	$$(-1)^nF^n(s)$$
$$\frac{1}{t}f(t)$$	$$\int_s^\infty F(u)du$$	$$\int_0^t f(v)dv$$	$$\frac{F(s)}{s}$$
$$\int_0^tf(t-\tau)g(\tau)d\tau$$	$$F(s)G(s)$$	$$f(t+T)=f(t)$$	$$\frac{\int_0^Te^{-st}f(t)dt}{1-e^{-sT}}$$
$$f'(t)$$	$$sF(s)-f(0)$$	$$f''(t)$$	$$s^2F(s)-sf(0)-f'(0)$$